You are probably talking about Elatior (hiemalis/rieger) begonias.

Most of these are treated as seasonal plants - buy them in bloom and when they start going downhill, throw them away.

I'm sure you can keep them for more than one season but the care may be more than it is worth. Good light, regular watering and fertilizing, removal of old leaves, etc should keep it going for longer than a few months.

Here are some websites that may be helpful.

http://hortchat.com/info/grow-rieger-begonia

http://www.ehow.com/info_8188772_elatior-begonia-care.html

http://homeguides.sfgate.com/elatior-begonia-care-26081.html

http://www.calyxflowers.com/Floral-Library/Content/Elatior-Begonia.aspx

The above links, clickable:
http://hortchat.com/info/grow-rieger-begonia
http://www.ehow.com/info_8188772_elatior-begonia-care.html
http://homeguides.sfgate.com/elatior-begonia-care-26081.html
http://www.calyxflowers.com/Floral-Library/Content/Elatior-Begonia.aspx

I was going to add a link to the Elatior section at ABS, but there isn't one. Wonder why?

Probably because most members don't grow them?

Thanks for making those into links (I was too lazy to do it).

Thanks to all who posted here for me. The begonias are doing ok so far. Maybe they needed more time to adjust to a new environment, and/or I bought ones previously that weren't adjusting well or were already diseased...

Yes, some plants have problems adjusting from a nursery to the home environment. Light levels, watering routines, warmth, air movement, etc all are big changes for the new plant. I find the potting mix is often the difference that makes or breaks a new plant.

Often the best solution is to change the potting mix ASAP since the way you water versus the way the plants were watered at the nursery may affect the plant greatly. I find a lot of begonias are planted in a mix of too much peat and watering must be with a very light hand else the mix will stay wet much too long which may lead to rot and disease.